angles between crystallographic axes if in monoclinic, or triclinic
systems.

Most of what you will do in lab or on exams will involve
crystallographic calculations in the orthorhombic, tetragonal, hexagonal,
or isometric systems, where the axes angles are fixed. Note that you will
always be given enough information to solve the problem. Some of the
problems you might be expected to solve involve determination of Miller
Indices from the ρ
and φ
angles, to determine the ρ
and φ
angles for faces of mineral with known axial ratios, or to determine axial
ratios of minerals that have faces of known ρ
and φ
angles.

First let's review some of the things we know about Miller indices and ρ
and φ angles.
In the orthorhombic, tetragonal, or isometric systems.

If h is 0, i.e. (0kl) is the Miller Index, then the face is parallel
to the a crystallographic axis.

So, φ =
0o or φ
= 180o

If k is 0 i.e. (h0l) is the Miller Index, then the face is parallel
to the b crystallographic axis.

So, φ
= 90o or φ
= 270o

If l is 0, i.e. (hk0) is the Miller Index, then the face is
parallel to the c crystallographic axis.

So, ρ
= 90o

If h & k are 0,
i.e. (00l) is the Miller Index, then the face
is perpendicular to the c axis

So, ρ
= 0o or ρ
= 180o, and φ
is undefined.

If h & l are 0, i.e.
(0k0) is the Miller Index, then the face
is perpendicular to the b axis.

So, ρ
= 90o and φ
= 0o or φ
= 180o.

If l and k are 0, i.e.
(h00) is the Miller Index, then the face is
perpendicular to the a axis.

So, ρ
= 90o and φ
= 90o or φ
= 270o.

Now let's do some examples

Given Miller indices and

ρ
and φ
angles for crystal faces that, in combination, intersect all crystallographic axes, calculate
the axial ratios of the mineral.

Faces
ρφ(110)
90o 45o(011) 70o
0o

For the (110) we note that it does not intersect the c axis,
so we can look at it in the two dimensional plane containing the a
and baxes, as shown in the drawing below. From this face we
should be able to determine the a/b axial ratio.
Since the φ angle is the
angle between the normal to the face and the b axis, by similar
triangles we know that the φ angle
also occurs between the a axis and the face.

Thus, we can write:

tan 45o = 1b/1a

then

1a/1b = 1/tan 45o= 1

So, a/b: b/b = 1: 1

We next note that the (011) intersects the b and c axes
only, so we can examine this face in the plane containing only b
and c, as shown below. From this drawing we can obtain the
c/b axial ratio. Since the ρ angle is
the angle between the pole to the face and the c axis, again by similar
triangles we know that the ρ angle also
occurs between the b axis and the (011) face.

The face (120) does not intersect the c axis, so
we can look at this face in the plane containing only the a and b
axes. We must also remember that Miller Indices represent the
inverse of the intercepts, so the face (120) intersects the a
axis at twice the number of unit lengths that it intersects the b
axis.
Since the φ angle is the
angle between the normal to the face and the b axis, by similar
triangles we know that the φ angle
also occurs between the a axis and the (120) face.

Then we can write:

tan 70o = 1b/2a

a/b = 1/2tan70o

1a/1b = 0.18199

We next note that the (011) intersects the b and c axes
only, so we can examine this face in the plane containing only b
and c, as shown below. From this drawing we can obtain the
c/b axial ratio. Since the ρ angle is
the angle between the face and the c axis, again by similar
triangles we know that the ρ angle also
occurs between the b axis and the (011) face.

For this face we can determine that

tan 32o = 1c/1b

1c/1b = 0.6248

so, a : b : c = 0.18199 : 1 : 0.6248, and the mineral is
orthorhombic

c. Faces

ρφ(311) 24o 33o

This face is more complicated because it intersects all
three axes.

We first attempt to draw a 3-dimensional view of this face.
Notice that the φ angle is
measured in the horizontal plane that includes the a and b
axes. The ρ angle is measured in a vertical
plane that includes the c axis and the line normal to the face in
the a - b horizontal plane, and is measured between the c
axis and a line normal to the face.

Note also, that for the (311) face, the intercept on the a axis
is 1/3 what it is on the b and c axes, because the Miller
Index is the inverse of the intercepts..

We can determine the a/b part of the axial ratio by looking at the
projection of this face in the a - b plane.

1b/(1/3)a = tan 33o

1a/3b = 1/tan 33

1a/1b = 3/tan 33

1a/1b = 4.6196

In order to determine the length of the c axis, we need to
know the length of the line labeled t, because this line forms the base
of the triangle in which the ρ angle is
measured. The length of the line t is:

t/b = cos 33o

t/b = 0.8397

t = 0.8397 b

We can now use this to determine the c/b axial ratio.

1c/t = tan 24o

1c = 0.8387 b tan 24o

c/b = 0.3747

Thus, the axial ratio for this mineral is 4.6196 : 1 : 0.3747

Now we'll look at an example where we are given the axial
ratio of the mineral and asked to calculate the

ρ
and φ
angles for the faces.

Given the axial ratio for a mineral is 1 : 1 : 5.0, what are

ρ
and φ
for the face (111).

We first attempt to draw a 3-dimensional view of this face. Notice
that again the φ angle is measured in
the horizontal plane that includes the a and b axes. The ρ angle is measured in a vertical
plane that includes the c axis and the line normal to the face in
the a - b horizontal plane, and is measured between the c
axis and a line normal to the face.

In this case the intercept on the all three axes is 1.

Since the φ angle for this
face is measured in the horizontal a - b plane, we can
draw the plane containing only the aand b axes to
determine the angle.

Since the axial ratio tells us that the relative lengths of the a
and b axes are equal

tan φ = 1b/1b = 1

φ = 45o

In order to determine the ρ angle, we need
to know the length of the line labeled t, because this line forms the
base of the triangle in which the ρ angle is
measured. The length of the line t is:

t/b = cos 45o

t = 0.7071 b

Now we can determine the angle by drawing the plane that
includes the c axis and the line t. In this plane we can
let the length of the c = 5b, from the axial ratio. Then:

tan ρ = 5b/t

tan ρ = 5b/0.7071b

tan ρ = 5/0.7071

tan ρ = 7.071

ρ = arctan (7.071) = 81.95o

So for the (111) face in this crystal ρ
=81.95o and φ = 45o.

Given the tetragonal crystal shown below and the following
information:

For the face (101)

ρ
= 70oφ
= 90o

What is the axial ratio for this crystal?

What are the Miller Indices for the faces

labeled
(0kl) and (h0l), given that both of these faces have ρ
= 53.9478o?

To find the axial ratio we note that we can use the face (101) and
draw this in plane of the a and c axes (since the face
doesn't intersect the b axis. Then, we can determine that

tan

ρ
= 1c/1a

so, tan 70o = c/a = 2.7475

and the axial ratio is: 1 :1 : 2.7475

To find the Miller indices of the face (h0l) we proceed as follows:

Since all we need is the relative lengths of intersection to calculate
the parameters, we can assume one of the lengths = 1, i.e. 1a. This is
the same as moving the face parallel to itself so that it intersects
the a axis at a unit length of 1. The face intersects the c axis at
2.7475x, where the 2.7475 value is the length of the c axis relative to
the unit length of the a axis. Then

tan (53.9478) = 2.7475x/1, and thus, x = 0.5

The parameters for this face are then:

1,

¥,
1/2, which can be inverted to give the Miller Index - (102).

Since the mineral is tetragonal, the face labeled (0hl) would have the Miller Index (012).